Preserver Problems On Generalized Inverses Of Matrices Over Field

Preserving invariant problems over matrix space is an active research area inmatrix theory. This paper investigated the linear preserver problems when the invariant was the generalized inverses of matrices. Let F be a field, M_n (F)the n×n full matrix algebra over F , and f denote the linear map over M_n(F).The generalized inverses of matrices and the preserver problems about the generalized inverses were outlined. The definition, quality of the generalized inverses and the basic knowledge of the linear map were given in this paper.However, when the characteristic of the base field or the base ring was 2, as to the preserving of the generalized inverses, the results were less. As to the characteristic 2, because of higher difficulty no results on the addition maps were obtained and more the discussed linear maps were invertible plus more conditions on the base fields. This paper removed the assumption that f was a linear bijection and gave the forms of linear maps from M_n (F) to itself preservinggroup inverses of matrices when the characteristic of the field was 2.Using the results of preserving idempotent matrices, the forms of linear maps from M_n(F) to itself preserving commutative {1}-inverses of matrices werecharacterized when the characteristic of the field was not 2. The conclusions were given in the theorem form, and the detail proof was given.